FIG. 1 shows a generalized Chua's circuit that has a parallel combination of a capacitor C2, a Chua's diode NR1 connected to an L-C tank circuit that includes a second capacitor C1 connected in parallel with a serially connected inductor L1. In addition, a first voltage V1 is defined across capacitor C1 and a second voltage V2 is defined across capacitor C2 with the positive orientation of voltages V1 and V2 both being at the positive end of the corresponding capacitor. Finally, there is a linear resistor R1 connected between the positive terminals of each capacitor.
In a typical configuration, the non-linear curve is piece-wise linear with symmetrical slope discontinuities around the current axis. It satisfies the following equation: IR=GaVR+(½) (Ga−Gb) {|vR+BP|−|vR−Bp|}, where Ga and Gb are the slopes of respective linear portions of the piece-wise linear current/voltage curve characterizing the nonlinear resistor, and Bp is the absolute value of the two voltage points at which discontinuities in the current/voltage curve lie, as shown in FIG. 2. The circuit has a circuit driving subsystem, an L-C tank circuit, and a response subsystem with a parallel combination of a capacitor and a non-linear resistor interconnected through a resistor.
By choosing values of R, L, C1 and C2, the circuit can operate in different operating regions, for example, the double scroll region. Herein, a Chua's circuit can be made that will oscillate chaotically or quasi-periodically. Given a specified physical configuration, a specified initial state specified by V1, V2 and IL, the voltage across the capacitors C1 and C2, and the current through the inductor L, the evolution of the Chua's circuit is deterministic, but chaotic. That is, any Chua's circuit with the same physical parameters and initial conditions will follow the same course of states over time, and this course will repeat itself over a very long period. However, to an observer it looks mainly like noise. Also, such a system's trajectory is sensitive to initial conditions. In addition, the power spectral density function is spread over a wide range of frequencies, with the peak frequency of the fundamental being governed by the L-C tank circuit.
Due to its simple circuitry and ability to demonstrate most well known routes to chaos, a Chua's Circuit and Chua's Oscillator (CC/CO) are an active topic of research in the study of non-linear dynamic circuits and systems. Recently, there has been an increasing interest in designing inductor-less Chua's circuits and Chua's oscillators (CC/CO). Moreover, due to its advantages, the attainability of all the three state variables of a CC/CO is also attracting the designers. Simultaneously, literature is also witnessing the shift of analog integrated circuit designs from voltage mode processing to current-mode processing (CMP). A number of chaotic circuits have been implemented using current-mode active building blocks. The Dual-Output Current Conveyor (DO-CCII) is also emerging as a versatile block to implement current-mode circuits.
To improve the performance of the circuit, different researchers have done extensive research. A paper by Kennedy M. P. (Kennedy M. P., ‘Robust op-amp realization of Chua's circuit’, Frequenz, 1992, 46, pp. 66–80) suggests a Chua's circuit using off the shelf components. Further, in another research paper by Torres, L. A. B. and Aguirre, L. A. (Torres, L. A. B. and Aguirre, L. A., Inductorless Chua's circuit, Electron. Lett., 2000, 36, (23), pp. 1915–1916) report a Chua's circuit using an operational amplifier to generate a Chua's oscillation at a very low frequency for bio-medical operations.
In a design proposed by Morgul, 0. (Morgul, O., ‘Inductorless realization of Chua's oscillator’, Electron. Lett., 1995, 31, pp. 1403–1404) synthetic inductors using op-amps were used along with the operational amplifier, thereby making the design suitable for monolithic implementation.
Senani R. and Gupta S. S. (Senani R. and Gupta S. S., ‘Implementation of Chua's chaotic circuit using current feedback op-amps’, Electron. Lett., 1998, 34, (9), pp. 829–830) proposed a Chua's circuit using a Current Feedback Operational Amplifier (CFOA), thus making available the third state variable through the inductor (iL).
In another architecture designed by Elwakil A. S. and Kennedy M. P. (Elwakil A. S. and Kennedy M. P., ‘Improved implementation of Chua's chaotic oscillator using current-feedback op-amp.’, IEEE Trans. CAS-I, 2000, 47, (1), pp. 76–79) the CFOA was efficiently used in the Chua's circuit to provide a higher bandwidth of the chaotic signal with a buffered output of one state variable.
All the above architectures individually provide advantages of a Chua's circuit, but so far, there is no circuit that simultaneously provides: current mode operation; use of minimum grounded passive elements; availability of all the state variables; availability of two state variables in the form of current which can be used to further generate other complex chaotic circuits; a circuit free from passive component matching; use of lesser active components as compared to that represented by FIG. 2; and generation of a reduced hardware higher order chaotic circuit (also called a hyper-chaotic circuit) using one of the available currents mentioned above. Thus, there is a need to develop a circuit that can provide all of these simultaneously.
A Chua's circuit, apart from being a device for demonstrating, studying and modeling a chaotic real world system, has been proposed to generate a hyper-chaotic circuit (T. Kapitaniak, L. O. Chua and G. Zhong, ‘Experimental Hyperchaos in Coupled Chua's Circuits’, IEEE Trans. CAS-I, Vol 41, No.7, July 1994). Referring to FIG. 3, it has been shown in (T. Kapitaniak, L. O. Chua and G. Zhong, ‘Experimental Hyperchaos in Coupled Chua's Circuits’, IEEE Trans. CAS-I, Vol 41, No.7, July 1994) that a Chua's circuit (100) may be coupled to a similar Chua's circuit (101) such that the non-grounded terminal of the inductor is connected to the input of a voltage buffer (10) whose output is connected to one of the terminals of a controlling resistor (11), and another end of the controlling resistor is connected to the non-grounded terminal of another Chua's circuit (101). The system coupling is achieved by interconnecting n Chua's circuit in the fashion described above, and the last Chua's circuit of the ring is either connected to the first Chua's circuit (100) or is left open. This type of coupling results in the following cases:
For the value of the controlling resistor greater than a specific value (called a threshold value), all the Chua's circuits will synchronize with each other. This value of the controlling resistor depends on the number of Chua's circuits used in the chain. During this state, the curve between the state variable of one Chua's circuit to the corresponding state variable in another Chua's circuit of the chain will be a straight line.
If the value of the controlling resistor is less than the specific value (called a threshold value), the Chua's circuits will loose synchronization with each other and the system will undergo a state of hyper-chaos, which is more sensitive to the initial condition. This value of the controlling resistor depends on the number of Chua's circuits used in the chain. During this state the curve between the state variable of one Chua's circuit to the corresponding state variable in another Chua's circuit of the chain will not be a straight line.
With such a proposal, a monolithic implementation of the hyper-chaotic circuit can be achieved by using any of the above-mentioned variations of the Chua's circuit (i.e., prior art Chua's circuits can be used to generate the hyper-chaotic circuit using the above described scheme). However, the use of any of these circuits will not only add to the disadvantage of those proposed Chua's circuits, but also one voltage buffer and one floating resistor per coupling is required, which in turn, makes the final circuit bulky and inefficient in terms of power consumption.